A well known theorem of Bochner says that the first Betti number of compact manifolds with positive Ricci curvature vanishes. More generally, D. Meyer used the Bochner technique to show that manifolds with positive curvature operators are rational homology spheres. In this talk I will explain that this is more generally the case for manifolds with $\lceil \frac{n}{2} \rceil$-positive curvature operators. We will see that this is a consequence of a general vanishing and estimation theorem for the $p$-th Betti number for manifolds with a lower bound on the average of the lowest $(n-p)$ eigenvalues of the curvature operator. This talk is based on joint work with Peter Petersen.